A letter is known to have come either from $\text{LONDON}$ or $\text{CLIFTON};$ on the postmark only the two consecutive letters $ON$ are ellegible. The probability that it came from $\text{LONDON}$ is:
A$\frac{5}{17}$
B$\frac{12}{17}$
C$\frac{17}{30}$
D$\frac{3}{5}$
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B$\frac{12}{17}$
We define the following events:
$A_1:$ Selecting a pair of consecutive letters from the word $\text{LONDON}$
$A_2:$ Selecting a pair of consecutive letters from the word $\text{CLIFTON}$
$E:$ Selecting a pair of letters $ON$
Then ${\text{P(A}_1∩\text{E})=\frac52},$ as there are $5$ pairs of consecutive letters out of which $2$ are $ON.$
$\text{P(A}_2∩\text{E})=\frac61,$ as there are $6$ pairs of consecutive letters of which $1$ is $ON.$
So, required probability $\text{P}=\Big(\frac{\text{A}_1}{\text{E}}\Big)$
$\Rightarrow\Big(\frac{\text{A}_1}{\text{E}}\Big)=\frac{\text{P}(\text{A}_1\cap\text{E})}{\text{P}(\text{A}_1\cap\text{E}) + \text{P}(\text{A}_1\cap\text{E})}=\frac{\frac25}{\frac25+\frac16}=\frac{12}{17}$
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